What is the sum of the interior angles of a pentagon? Circle your answer. And the answers are 360 degrees, 180 degrees, 540 degrees, and 720 degrees.
Well, if we think about a pentagon, a pentagon is a five-sided shape. And what we want to do is find the sum of the interior angles. And I’ve marked the interior angles of the pentagon on my diagram.
Now there are a couple of ways that we can do this. The first way is to divide our pentagon into triangles. So I’ve started at one vertex. And then I’ve drawn a line to other vertices that create triangles. And as you can see, I’ve made three triangles.
And we know that the interior angles of a triangle add up to 180 degrees. So therefore, the sum of the interior angles of our pentagon is gonna be equal to three, because there are three triangles, multiplied by 180, because that’s the interior angles of a triangle, which would give us the answer 540 degrees.
And we could’ve worked this out using column multiplication. So first of all, we’d have three multiplied by zero, which is just zero. Then we do three multiplied by eight, which is 24. So we put four in the tens column, carry the two into the hundreds column. Then we’ve got three multiplied by one, which is three, and then add the twos that we carried makes five. And it gives us our 540.
Okay, so great, that’s the first method that I wanted to show you. And we have a formula that will represent what we’ve just done for us. And that formula is that the sum of the interior angles of a polygon is equal to 180 then multiplied by 𝑛 minus two, where 𝑛 is the number of sides, cause if we look at what we’ve done, we’d have 180 multiplied by five minus two, cause the pentagon has five sides. Well, that’s 180 multiplied by three, which is what we did and what we showed with the triangles.
Well, the second method you could use to find out the sum of the interior angles is to think about the exterior angles of our pentagon. And to calculate the exterior angle of a polygon, we have a formula again. And this one is 360 divided by 𝑛, where 𝑛 is the number of sides of the polygon.
So therefore, if we divided 360 by five, cause five is the number of sides of our polygon, we get an exterior angle of 72 degrees. And again, we could’ve used the written method to work this out. So if we see how many fives go into 360 using the bus stop method, fives in three don’t go. So we carry the three. Then we do fives into 36. And this goes seven times with one remainder. Then we do fives into 10. This goes twice. So we get our 72.
Okay great, so we’ve got our exterior angle. But how does this help us? And then to work out our interior angle, what we can do is do 180 minus 72. And that’s because we know that an exterior angle plus an interior angle is equal to 180 degrees. And this will give us an interior angle of 108 degrees.
This is however assuming that we’ve got a regular pentagon. It doesn’t say that in the question. But by assuming that, we can calculate the sum of the interior angles of a pentagon because then it wouldn’t matter if it is regular or irregular because the sum is always the same.
And therefore, the sum of the interior angles is gonna be equal to 108 multiplied by five. And that’s because there’s five interior angles. So we do five multiplied by 100 is 500 and five multiplied by eight is 40. So we get 540 degrees. And this agrees with our first method. So therefore, we can confirm that the sum of the interior angles of a pentagon is 540 degrees.